A structural kinetics model for thixotropy

doi:10.1016/j.jnnfm.2006.06.002

Journal of Non-Newtonian Fluid Mechanics

Volume 139, Issues 1–2, 15 November 2006, Pages 21-30

https://doi.org/10.1016/j.jnnfm.2006.06.002 Get rights and content

Abstract

A general structural kinetics model is presented to describe the flow behaviour of thixotropic systems based on inelastic suspending media. The total stress is divided in structure-dependent elastic and viscous contributions. The kinetic equation for the structure parameter takes into account the effect of shear on structure breakdown and build-up, as well as the effect of Brownian motion on build-up. The relaxation and deformation of the flocs is also included. Both the kinetic and the relaxation equations contain a distribution of time constants. The predictions of this model, as well as those of two representative models from the literature, are compared with experimental data using an objective method for parameter estimation. Model validation is based on stress transients resulting from sudden changes in shear rate, including both structure build-up and breakdown. The predictions of the viscous and elastic components are evaluated separately using stress jump experiments with steady and non-steady state starting conditions.

Introduction

Many technical and commercial products consist of weakly flocculated dispersions that display thixotropic behaviour. Thixotropy implies that the viscosity gradually decreases in time when the shear rate is suddenly increased. This time effect should be reversible and consequently the viscosity should again increase in time when the shear rate is subsequently lowered. Mewis [19], and more recently Barnes [2], have presented extensive reviews on the subject. The time effects in thixotropic materials are associated with the gradual breakdown and build-up of the microstructure. Viscoelastic effects such as stress relaxation and the first normal stress difference are often marginal phenomena in these systems and are therefore neglected.

The numerous thixotropic models that have been proposed are based on three different approaches [19]: a continuum mechanics approach, a microstructural and a structural kinetics one. Within the framework of rational continuum mechanics, inelastic thixotropy models, e.g. [20], [29], [31], are often generated by introducing a time-dependent yield surface and/or a time-dependent viscosity function into existing models such as the Bingham or the Reiner–Rivlin equations. The time function can be written as a kind of memory function. The model parameters of these phenomenological models are, however, not associated with the physical processes responsible for the structural changes and the details of the model are often specified in a rather arbitrary fashion. The microstructural models, on the other hand, attempt to derive the rheological behaviour of thixotropic systems from an accurate description of the basic physical mechanisms that govern the reversible build-up and breakdown of flocs [10], [26], [3]. To generate such models the underlying physical model equations should be available and the necessary parameters, here related to the interparticle interaction forces, should be known. This is often not the case for real systems. Only fragmentary validation of such models can be found in the literature [10], [13].

The third group consists of the structural kinetics models. A general framework for this modelling approach has been presented by Cheng and Evans[6]. In these models, the instantaneous structure is characterized by means of a scalar structural parameter, e.g. $λ$ (0 $\leq λ \leq 1$ ). The completely build-up and completely broken down structures are usually represented by a value for $λ$ of, respectively, 1 and 0. The first equation of a structural kinetics model is a constitutive equation, relating the instantaneous shear stress $σ$ to the instantaneous values of $λ$ and the shear rate $\dot{γ}$ . Mainly two types of constitutive equations have been used for thixotropic materials with inelastic media. In the first one the variable microstructure is assumed to cause a variable degree of plasticity, expressed by a yield stress that depends on the shear history [15], [34], [32], [24]. The second class of models assumes the microstructure to be deformable, which causes a viscoelastic contribution to the stress. In this case the basic constitute equations are nonlinear variants of the usual viscoelastic equations such as the Maxwell [17], [27], [8] or the Jeffrey models [38], [28].

Due to the wide range and the complex nature of the microstructures that can be encountered in thixotropic systems, the structural kinetics approach might be more suited for a general thixotropy model than the microstructural approach. A large number of structural kinetics models have been proposed in the literature. Their evaluation, however, has mostly been limited to either a qualitatitive demonstration of their potential or to applying them to limited data sets. More substantial quantitative model assessments are extremely scarce in the literature [1], [38]. Recently [14], it has been shown that the existing structural kinetics models display some generic shortcomings. Here, a new structural kinetics model is proposed that overcomes some of these problems. It will be evaluated with a systematic set of transient experiments resulting from sudden changes in shear rate. Both structure breakdown and structure recovery will be considered. Further model evaluation is in progress and will be published elsewhere.

Section snippets

Materials

Two different thixotropic systems will be used to validate the proposed structural kinetics model. The first system is based on fumed silica particles (Aerosil R972, provided by Degussa, Germany). These particles have been dispersed in a mixture of a highly refined paraffin oil (Riedel-de Haën 18512) and a low molecular weight poly(isobutylene) (PIB, Oppanol B3 from BASF). Details concerning the function of the different components in this model system as well as the specific conditions

Proposed model

As is common practice for structural kinetics models, only a one-dimensional version of the new model will be considered. The shear stress $σ$ is divided in a particle ( $σ_{p}$ ) and a medium ( $σ_{m}$ ) contribution: $σ (λ, \dot{γ}) = σ_{p} (λ, \dot{γ}) + σ_{m} (\dot{γ})$ where $\dot{γ}$ is the shear rate and $λ$ a scalar parameter that expresses the degree of structure and ranges from 0 to 1. The particle contribution in this model is written as the sum of an elastic ( $σ_{p}^{el}$ ) and a viscous, hydrodynamic, contribution ( $σ_{p}^{vis}$ ): $σ (λ, \dot{γ}) = σ_{p}^{el} (λ, \dot{γ}) + σ_{p}^{vis} (λ,$

Parameter estimation

Three of the eight model parameters can be determined directly from specific experiments. The value of the apparent yield stress, $σ_{y, 0}$ , is obtained by extrapolating (linear regression) the steady state stress at low shear rates to shear rate zero. The value of $γ_{c}$ is calculated from the ratio of $σ_{y, 0}$ over the shear modulus $G_{0}$ . The latter is identified with the low frequency plateau value of the storage modulus $G^{'} (ω)$ , measured after a sufficiently long rest period. The five remaining parameters,

Steady shear and stepwise changes in shear rate

The estimated parameter values for the fumed silica and carbon black dispersions, obtained using the procedure outlined above, are presented in Table 1. Confidence intervals (95%) have been calculated [4] for the different parameters and have been included in the table. The low level of uncertainty on the estimated parameter values demonstrates the quality of the parameter estimations.

Model calculations and experimental data for the steady state flow curve for both dispersions are shown in Fig.

Conclusions

A general structural kinetics model for thixotropic materials in Newtonian media is proposed. The stress contains structure-dependent elastic and viscous contributions. By means of a differential evolution equation for the elastic aggregate strain, stress-controlled relaxation and stretching of the aggregates is taken into account. The kinetic equation for the structure parameter contains terms for shear dependent breakdown, shear-induced aggregation and Brownian build-up of aggregates. The

Acknowledgements

The silica particles were kindly provided by Degussa (Germany). The authors are grateful to J.F. Van Impe and co-workers for their input during the construction of the identification routine.

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View full text

A structural kinetics model for thixotropy

Abstract

Introduction

Section snippets

Materials

Proposed model

Parameter estimation

Steady shear and stepwise changes in shear rate

Conclusions

Acknowledgements

J. Non-Newtonian Fluid Mech.

Chem. Eng. Sci.

Int. J. Food Microbiol.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

Modelling thixotropy using a novel structural kinetics approach: basis and application to a solution of iota carrageenan

J. Texture Stud.

On the use of stretched exponential functions for both linear viscoelastic creep and stress relaxation

Rheol. Acta

Phenomenological characterization of the rheological behaviour of inelastic reversible thixotropic and antithixotropic fluids

Br. J. Appl. Phys.

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J. Phys. D-Appl. Phys.

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